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To build the example, run in a Terminal:
cd /path-to-example/
cmake .
make
***********************************************************************************************************************
Example of use of RIPS:
Computation of the persistent homology with Z/2Z coefficients of the Rips complex on points
sampling a Klein bottle:
./rips_persistence ../../data/points/Kl.txt -r 0.25 -d 3 -p 2 -m 100
output:
210 0 0 inf
210 1 0.0702103 inf
2 1 0.0702103 inf
2 2 0.159992 inf
Every line is of this format: p1*...*pr dim b d
where
p1*...*pr is the product of prime numbers pi such that the homology feature exists in homology with Z/piZ coefficients.
dim is the dimension of the homological feature,
b and d are respectively the birth and death of the feature and
with Z/3Z coefficients:
./rips_persistence ../../data/points/Kl.txt -r 0.25 -d 3 -p 3 -m 100
output:
3 0 0 inf
3 1 0.0702103 inf
and the computation with Z/2Z and Z/3Z coefficients simultaneously:
./rips_multifield_persistence ../../data/points/Kl.txt -r 0.25 -d 3 -p 2 -q 3 -m 100
output:
6 0 0 inf
6 1 0.0702103 inf
2 1 0.0702103 inf
2 2 0.159992 inf
and finally the computation with all Z/pZ for 2 <= p <= 71 (20 first prime numbers):
./rips_multifield_persistence ../../data/points/Kl.txt -r 0.25 -d 3 -p 2 -q 71 -m 100
output:
557940830126698960967415390 0 0 inf
557940830126698960967415390 1 0.0702103 inf
2 1 0.0702103 inf
2 2 0.159992 inf
***********************************************************************************************************************
Example of use of ALPHA:
For a more verbose mode, please run cmake with option "DEBUG_TRACES=TRUE" and recompile the programs.
1) 3D special case
------------------
Computation of the persistent homology with Z/2Z coefficients of the alpha complex on points
sampling a torus 3D:
./alpha_complex_3d_persistence ../../data/points/tore3D_300.off 2 0.45
output:
Simplex_tree dim: 3
2 0 0 inf
2 1 0.0682162 1.0001
2 1 0.0934117 1.00003
2 2 0.56444 1.03938
Here we retrieve expected Betti numbers on a tore 3D:
Betti numbers[0] = 1
Betti numbers[1] = 2
Betti numbers[2] = 1
N.B.: - alpha_complex_3d_persistence accepts only OFF files in 3D dimension.
- filtration values are alpha square values
2) d-Dimension case
-------------------
Computation of the persistent homology with Z/2Z coefficients of the alpha complex on points
sampling a torus 3D:
./alpha_complex_persistence -r 32 -p 2 -m 0.45 ../../data/points/tore3D_300.off
output:
Alpha complex is of dimension 3 - 9273 simplices - 300 vertices.
Simplex_tree dim: 3
2 0 0 inf
2 1 0.0682162 1.0001
2 1 0.0934117 1.00003
2 2 0.56444 1.03938
Here we retrieve expected Betti numbers on a tore 3D:
Betti numbers[0] = 1
Betti numbers[1] = 2
Betti numbers[2] = 1
N.B.: - alpha_complex_persistence accepts OFF files in d-Dimension.
- filtration values are alpha square values
3) 3D periodic special case
---------------------------
./periodic_alpha_complex_3d_persistence ../../data/points/grid_10_10_10_in_0_1.off 3 1.0
output:
Periodic Delaunay computed.
Simplex_tree dim: 3
3 0 0 inf
3 1 0.0025 inf
3 1 0.0025 inf
3 1 0.0025 inf
3 2 0.005 inf
3 2 0.005 inf
3 2 0.005 inf
3 3 0.0075 inf
Here we retrieve expected Betti numbers on a tore 3D:
Betti numbers[0] = 1
Betti numbers[1] = 3
Betti numbers[2] = 3
Betti numbers[3] = 1
N.B.: - periodic_alpha_complex_3d_persistence accepts only OFF files in 3D dimension. In this example, the periodic cube
is hard coded to { x = [0,1]; y = [0,1]; z = [0,1] }
- filtration values are alpha square values
***********************************************************************************************************************
Example of use of PLAIN HOMOLOGY:
This example computes the plain homology of the following simplicial complex without filtration values:
/* Complex to build. */
/* 1 3 */
/* o---o */
/* /X\ / */
/* o---o o */
/* 2 0 4 */
./plain_homology
output:
2 0 0 inf
2 0 0 inf
2 1 0 inf
Here we retrieve the 2 entities {0,1,2,3} and {4} (Betti numbers[0] = 2) and the hole in {0,1,3} (Betti numbers[1] = 1)
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